institute of statistics rwth aachen university · 2008-11-20 · acceptance sampling is concerned...

Preprint Series, Institute of Statistics, RWTH Aachen University

SAMPLING INSPECTION BY VARIABLES: NONPARAMETRIC SETTING

Ansgar Steland and Henryk Zahle

Institute of Statistics

RWTH Aachen University

[email protected]

and

Department of Mathematics

Technical University Dortmund

[email protected]

Abstract A classic statistical problem is the optimal construction ofsampling plans to accept or reject a lot based on a small sample. Wepropose a new asymptotically optimal solution for the acceptance sam-pling by variables setting where we allow for an arbitrary unknown un-derlying distribution. In the course of this, we assume that additionalsampling information is available, which is often the case in real appli-cations. That information is given by additional measurements whichmay be affected by a calibration error. Our results show that, firstly,the proposed decision rule is asymptotically valid under fairly generalassumptions. Secondly, the estimated optimal sample size is asymptot-ically normal. Further, we illustrate our method by a real data analysisand we investigate to some extent its finite sample properties and thesharpness of our assumptions by simulations.

Primary 60F05, 62D05, 62K05; secondary 60F17, 62G30.Keywords: Acceptance sampling, Frechet and Hadamard derivative,

functional delta method, central limit theorem, empirical process.

1. INTRODUCTION

Acceptance sampling is concerned with the (optimal) construction of sampling plans to

accept or reject a possibly large lot based on a small sample. This can be based on quality

characteristics with two outcomes (sampling by attributes), or characteristics measured

on a metric scale (sampling by variables). The classical procedures for the latter problem

assume normally distributed observations, which is too restrictive for many applications.

Thus, in this article we consider the problem to construct asymptotically optimal sampling

plans for acceptance sampling by variables for an arbitrary unknown underlying distribu-

tion, when additional sampling information is available. That additional information is

given by additional measurements of the lot or a further random sample from the pro-

duction process, where these measurements may be affected by calibration errors. This

1

http://www.stochastik.rwth-aachen.de/

mailto:[email protected]

mailto:[email protected]

2 A. STELAND AND H. ZAHLE

situation arises often in real applications. E.g., suppliers to industry usually deliver in lots,

and these lots are checked using acceptance sampling rules. As long as the distribution

does not change, one may pool the data from these analyses. Our work is also motivated

by a specific quality control problem dealing with photovoltaic modules. Here, due to

technical reasons, the distribution of these measurements varies considerably from lot to

lot, and is usually non-normal. However, producers sometimes hand out their lot measure-

ments. Although these measurements are often affected by a (possibly advisedly arranged)

calibration error, they can be used to estimate the shape of the underlying distribution.

For details we refer to [7].

Denote by X1, . . . XN a quality characteristic from a random sample from the production

process. The common distribution function of the Xi’s will be denoted by F . We assume

throughout that the fourth moment of F exists, and let µ and σ2 denote the mean and

variance of F , respectively. Further assumptions will be given in the next section. Item

i is called conforming (non-defect), if Xi > τ for some τ ∈ R. Then the fraction of

non-conforming (defect) items is given by the probability

(1.1) p := E(µ,σ2)

[1

N

N∑i=1

1Xi≤τ

]= P(µ,σ2)[X1 ≤ τ ],

and is the common quantity to define the quality of a lot. Here and in the sequel 1A

denotes the indicator function of the event A. The lot should be accepted if p is smaller

than the acceptable quality level (AQL) pα and rejected if p is larger than the rejectable

quality level (RQL) pβ. The subscripts α and β will be explained below. pα and pβ have

to be settled by an agreement between the producer and the consumer of the shipment. In

any case we wisely assume pα < pβ.

The classic approach to the problem is to construct a decision rule such that the error

probabilities of a false acceptance and a false rejection of the lot are controlled. For an

overview see the monograph [10], further information can be found in, e.g., [2, 3] and in

references cited therein. For normally distributed observations the problem is straightfor-

ward and well known. For instance, sampling plans for known variance, unknown variance,

and average range, which are based on the Rao-Blackwell estimator of the fraction of non-

conforming items, have been developed in [8]. Further plans for non-normal distributions

have been studied, but only for certain parametric models where the form of the underlying

distribution is known and the proportion of non-conforming items is a simple function of

the parameters. For instance, [18] studied inspection by variables based on Burr’s distri-

SAMPLING INSPECTION BY VARIABLES: NONPARAMETRIC SETTING 3

bution. When the distribution is known up to a location parameter, the problem has been

studied by [4, 16]. In this paper, we propose a nonparametric solution for the general case.

It is based on nonparametric estimators of all unknown quantities appearing in the formu-

lae that specify the sampling plan (in [4, 16]) when the normality assumption is dropped.

We provide a rigorous asymptotic justification for the proposed decision rule, which relies

on the delta method for Hadamard differentiable statistical functionals and on a functional

central limit theorem for empirical processes with estimated parameters. Background on

the latter tools is given in [1, 12, 17], for instance.

The organization of the rest of the paper is as follows. In Section 2 the distributional

model and the proposed decision rule are carefully introduced, and the assumptions re-

quired by our asymptotic result are discussed. Section 3 provides the main result. We

prove an asymptotic representation of the operating characteristic, which justifies a simple

plug-in rule for the choice of the sample size. The asymptotic behavior of that estimated

sample size is also investigated in terms of a central limit theorem. Section 4 provides a real

data analysis and results from an extensive simulation study. We analyze real photovoltaic

measurements and illustrate our solution to the problem. The data analysis shows that

there is indeed need for nonparametric acceptance sampling procedures. Our simulations

provide valuable insight into the applicability of the method. The results indicate that,

firstly, the proposed method works reliable in many situations of practical interest. Sec-

ondly, it provides evidence that the fourth moment assumptions required by our theoretical

results can not be weakened.

The proofs of the main results are given in Section 5. Several appendices provide further

technical details and a brief review of the functional delta method used in our proofs.

2. ASSUMPTIONS, MODEL, AND THE DECISION RULE

Clearly, the true fraction of non-conforming items p, which equals the expectation of

the fraction of non-conforming items of the lot of size N , is unknown. Since an inspec-

tion of all lot items is usually not feasible, the consumer’s decision to accept or reject

the shipment has to be based on n < N i.i.d. control measurements, X ′1, . . . , X′n, which

are also distributed according to F . We aim at defining a suitable decision rule based on

X ′1, . . . , X′n to accept or reject the lot. The additional information is given by a further

i.i.d. random sample X∆1 , . . . , X

∆m of size m representing, for instance, a data sheet from

the producer. The values X∆i may be affected by a calibration error ∆ relative to the

control measurements. That is, we just require X∆i

d= Xj + ∆, for some ∆ ∈ R and all i, j.


We emphasize that the samples (X ′i) and (X∆i ) are not required to be independent, and

we assume 1 n m ≤ N . We will refer to the basic probability space as (Ω,F ,P(µ,σ2))

and assume that it is rich enough to host whole sequences (X ′i) and (X∆i ) as i.i.d. random

variables with distribution function F .

Concerning F , we do not assume any parametric form, but only require that F is a

member of the nonparametric location-scale family

F :=F(µ,σ2)(.) := G((.− µ)/σ) : (µ, σ2) ∈ R× (0,∞)

,

where G (= F(0,1)) is an arbitrary but unknown distribution function.

Before proceeding with the exposition of the statistical problem and the proposed solu-

tion, let us phrase and discuss our rather weak assumptions on the distribution function

G.

Assumption 2.1∫R xdG(x) = 0,

∫R x

2dG(x) = 1, and∫R x

4dG(x) <∞.

Assumption 2.2 G is continuously differentiable, and

(2.1)∥∥∥F(µ,σ2)+(v1,v2)(.)− F(µ,σ2)(.)− [F(µ,σ2)(.)](v1, v2)′

∥∥∥∞

= o(|v|)

for every fixed (µ, σ2) ∈ R× (0,∞), where (v1, v2)′ is the transpose of v = (v1, v2), and

(2.2) F(µ,σ2)(t) :=

(∂

∂µF(µ,σ2)(t),

∂

∂σ2F(µ,σ2)(t)

)= − 1

σ

(G′(t− µσ

),t− µ2σ2

G′(t− µσ

)).

Assumption 2.3 G is strictly increasing on [a, b], where a := supt ∈ R : G(t) = 0 and

b := inft ∈ R : G(t) = 1 with the conventions sup ∅ := −∞ and inf ∅ :=∞.

Here, ‖.‖∞ denotes the usual supremum norm. Assumption 2.3 implies in particular

that the map (µ, σ2) 7→ F(µ,σ2) is injective. Assumption (2.1) just means that the mapping

(µ, σ2) 7→ F(µ,σ2) (from R × (0,∞) to the space of all cadlag functions on R equipped

with ‖.‖∞) is Frechet differentiable at (µ, σ2) with Frechet derivative F(µ,σ2). The notion of

Frechet differentiability and the definition of cadlag functions can be found, for instance,

in [17, p.297 and p.257]. Assumption 2.2 is quite an abstract condition, and therefore we

mention a more transparent sufficient condition. If G is twice continuously differentiable

and satisfies

(2.3)∫

R|x| |G′′(x)|dx <∞,


then (2.1) holds, cf. Lemma A.1 in the Appendix A. With the help of this sufficient con-

dition one can easily check that for instance the normal distribution satisfies Assumption

2.2 (apart from Assumptions 2.1 and 2.3). Further examples for G satisfying (2.3) are all

twice continuously differentiable distribution functions with compact support or Pareto-

type tails κ|x|−γ (for x away from 0) with γ > 1 and suitable κ = κγ > 0.

To motivate our proposal, let us first consider the treatment of the acceptance by variables

problem when σ and G are known (this condition will be skipped below). From an intuitive

point of view, the lot should be accepted if and only if

(2.4)√nX ′n − τσ

> c,

where c is some suitable positive constant and X ′n is the sample mean of X ′1, . . . , X′n. The

free parameters n and c determine the sample size and the acceptance range, respectively.

The consumer is obviously interested in a small sample size n. This interest is however

foiled by the problem that a small n does not admit a safe decision. Intuitively, a small

sample size n cannot ensure that a decision rule keeps the probability of the type II error

(consumer risk) small. On the other hand, the producer will typically insist on a small

probability of the type I error (producer risk). Recall that the type I error paraphrases a

rejection of the lot although it is of high quality. In contrast, the type II error paraphrases

an acception of the shipment although it is of low quality. A fair decision rule should keep

both the type I error and the type II error small. That is, for 0 < β < α,

(2.5) p ≤ pα ⇒ prob [“acceptance”] ≥ α,

(2.6) p ≥ pβ ⇒ prob [“acceptance”] ≤ β.

The values α and β determine the error probabilities the contracting parties are willing to

accept. Hence, α and β fix the confidence level of the decision rule. In particular, they

should be part of the contract to supply the shipment.

The basic question is how small n may be in order to gurantee (2.5)-(2.6). Toward an

answer we note that the test statistic on the left-hand side of (2.4) can be written as the

sum of√n(µ − τ)/σ and an expression that is asymptotically standard normal. By (1.1)

we also have p = G((τ−µ)/σ), where G(t) = F (σt+µ) denotes the distribution function of


the standardized variable (X1− µ)/σ. Thus, if G is strictly increasing and n is reasonably

large, we have

(2.7) prob [“acceptance”] ≈ 1− Φ(c+√n G−1(p)

)with Φ the distribution function of the standard normal distribution and G−1(p) = inft ∈R : G(t) ≥ p the p-quantile of G. From (2.7) one easily deduces (cf. Appendix B)

the minimal sample size n along with the acceptance threshold c such that (2.5)-(2.6) is

ensured:

(2.8) n =

(Φ−1(1− α)− Φ−1(1− β)

G−1(pβ)−G−1(pα)

)2

(2.9) c = Φ−1(1− α)−√n G−1(pα).

Strictly speaking, (2.5)-(2.6) are ensured only approximately. Note that c in (2.9) remains

unchanged when α is replaced by β. This is a by-product of the derivation of (2.8)-(2.9)

in the Appendix B.

A natural way to handle the unknown quantities σ and G is to plug in appropriate

estimators. Here the additional sample X∆1 , . . . , X

∆m comes into play. The superscript

indicates that it may be shifted by a constant relative to the control measurements, but

the following estimators of σ and G are location invariant, i.e., robust against shifts.

(2.10) σm :=

(1

m− 1

m∑i=1

(X∆i − X∆

m

)2)1/2

,

(2.11) Gm(t) :=1

m

m∑i=1

1(−∞,t]((X∆

i − X∆m)/σm

)(t ∈ R).

Note that Gm is the empirical distribution function of the (empirically) standardized ran-

dom variables (X∆1 − X∆

m)/σm, . . . , (X∆m − X∆

m)/σm. We now replace the left-hand side of

(2.4) by

Tn :=√nX ′n − τσm

to obtain the following decision rule:

Rule 2.1 The lot will be accepted if and only if Tn > c.


Now intuition suggests to use the formulae (2.8) and (2.9) with G replaced by Gm. We

postpone a more detailed derivation and a rigorous justification of this idea to the next

section.

For that main result we need the following assumption that the sample size m is essen-

tially larger than n, i.e.,

(2.12) m = mn and limn→∞

n/m = 0.

This assumption is indeed crucial. If m and n grew at the same rate, i.e. if limn→∞ n/m

existed in (0,∞), one might still obtain versions of (2.8) and (2.9), but belike they would

depend on G which was pointless. For this obvious reason we impose (2.12), and, as

indicated in the introduction, that assumption is not restrictive for applications.

3. MAIN RESULT

Recall that our objective is to determine the acceptance range, i.e., the threshold c, and

the minimum sample size n of Rule 2.1 such that (2.5)-(2.6) is ensured. The specification

of (n, c) is given in (3.6)-(3.7) which, to some extent, is the main result of this article.

Toward the justification we first rewrite (2.5)-(2.6). To this end we introduce the operation

characteristic (or acceptance probability function)

(3.1) An,c(p) := P(µ,σ2)[Tn > c] (p ∈ [0, 1]).

Recalling p = G((τ−µ)/σ) and taking Assumption 2.3 into account we deduce a one-to-one

correspondence between p and µ = µ(p), so that the right-hand side of (3.1) can indeed be

seen as a function of p. Now we can rewrite (2.5)-(2.6) as

(3.2) p ≤ pα ⇒ An,c(p) ≥ α,

(3.3) p ≥ pβ ⇒ An,c(p) ≤ β.

As we did not restrict F to be any specific parametric distribution, we have no option but

require (3.2)-(3.3) only asymptotically (n→∞). The key for the formulae (3.6) and (3.7)

below is the following main theorem.

Theorem 3.1 Suppose (2.12) and Assumptions 2.1-2.3 hold. Then, for every p ∈ (0, 1),

there exists a sequence (δn(p)) of random variables converging in P(µ,σ2)-probability to 0

and satisfying for all c ∈ R,

(3.4) An,c(p) = P(µ,σ2)

[√nX ′n − µσ

+ δn(p) ≥ c+√n G−1

m (p)

].


The proof is postponed to Section 5. According to Theorem 3.1 we have for reasonably

large n the analogue of (2.7),

(3.5) An,c(p) ≈ 1− Φ(c+√n G−1

m (p)).

Indeed, the left-hand side of the event in (3.4) is asymptotically standard normal by the

central limit theorem and Slutsky’s lemma. Now we may proceed as in the Appendix B,

with G replaced by Gm, to obtain the (asymptotically) optimal sampling plan, i.e., (n, c)

with the minimal n satisfying (3.2)-(3.3):

(3.6) n = nm =

(Φ−1(1− α)− Φ−1(1− β)

G−1m (pβ)−G−1

m (pα)

)2

(3.7) c = cm = Φ−1(1− α)−√n G−1

m (pα).

Note that c in (3.7) remains unchanged when α is replaced by β. This is a by-product of

the considerations in the Appendix B. Our analysis suggests that the sample size n should

be chosen as the smallest integer being larger than the right-hand side of (3.6).

At first glance, formula (3.6) seems to be pointless. It provides a formula for n but the

right-hand side implicitly depends on n since we presupposed (2.12). Recall, however, that

there are many practical situations where X∆1 , . . . , X

∆m , and therefore Gm and σm, are a

priori known for a given “sufficiently large” m, but the shift ∆ is unknown. This is for

instance exactly the case with our motivating example where the producer hands out a

data sheet. We also emphasize that our asymptotic framework is by all means justified, at

least for a stringent choice of α and β. More precisely, if pα and pβ are fixed and if α 1

or β 0 then (Φ−1(1−α)−Φ−1(1− β)) approaches −∞, so that n in (3.6) tends to +∞since (G−1

m (pβ) − G−1m (pα)) remains bounded with high probability. Note that the latter

expression converges in probability to the constant (G−1(pβ) − G−1(pα)), as m → ∞, by

Lemma 5.2.

The following theorem provides an intuition of the asymptotic behavior (m→∞) of the

sample size n = nm specified by (3.6). The proof is postponed to Section 5.

Theorem 3.2 Let nm and n∞ be defined by the right-hand sides of (3.6) and (2.8),

respectively. Then, under P(µ,σ2),

(3.8)√m (nm − n∞)

d→ N(0, V (α, β, pα, pβ)),


as m→∞, where

(3.9) V (α, β, pα, pβ) := 4(Φ−1(1− α)− Φ−1(1− β))4

(G−1(pβ)−G−1(pα))6V (pα, pβ)

with

V (pα, pβ) :=Γ(G−1(pβ), G−1(pβ))

G′(G−1(pβ))2− 2 Γ(G−1(pα), G−1(pβ))

G′(G−1(pα))G′(G−1(pβ))+

Γ(G−1(pα), G−1(pα))

G′(G−1(pα))2.

Here Γ is a symmetric function on R× R defined by

Γ(s, t) := G(s)(1−G(t))− G(t)b(s)− G(s)b(t) + G(s)AG(t)′

for s ≤ t, where

G(t) := − (G′(t), (t/2)G′(t))

b(t) :=

E(0,1)

[X11(−∞,t](X1)

]E(0,1)

[(X2

1 − 1)1(−∞,t](X1)]

A :=

1 E(0,1)[X31 ]

E(0,1)[X31 ] E(0,1)[X

41 ]− 1

.The theorem shows that

√m (nm−n∞) is asymptotically normal. The good thing is that

we have specified the asymptotic variance explicitly. Unfortunately the formula is rather

complicated. We can see at least that the asymptotic variance is independent of (µ, σ2).

Moreover, the power six of the denominator on the right-hand side of (3.9) indicates that the

fluctuation of nm might be relatively strong even for large values of m. This phenomenon

can indeed be observed in the table given in Subsection 4.2. At this point we emphasize

that (G−1(pβ)−G−1(pα)) is typically smaller than 1, and that |Φ−1(1−α)−Φ−1(1−β)| istypically larger than 1. If, for example, 1− α = β = 0.05, AQL pα = 0.01, RQL pβ = 0.05

and G = Φ, then (G−1(pβ)−G−1(pα)) ≈ 0.6815 and |Φ−1(1− α)− Φ−1(1− β)| ≈ 3.2897.

4. EXAMPLE AND SIMULATIONS

In this section, we illustrate the proposed method by a real data example and study the

statistical properties by a Monte Carlo study. The latter addresses two issues. First, we

were interested in the finite sample behavior of the procedure when applied to realistic

sample sizes and distributional models of practical relevance. Second, we analyzed to

some extent whether the assumptions of our limit theorem are sharp. Here the fourth

moment assumption is of primary concern, because it rules out distributions with thick

tails. Recall that the thickness of the tails is often measured by the (empirical) kurtosis,

i.e., the (empirical) fourth moment after (empirical) standardization.


4.1. Example

Our work is motivated by a project with TUV Rheinland Immissionsschutz und En-

ergiesysteme GmbH, which offers quality control services for producers of photovoltaic

modules. We applied the procedure to real power measurements under so-called standard

conditions. The producer handed out a so-called flasher list of m := N = 500 measure-

ments with σm = 4.23, a subsample of a larger list. The Shapiro test for normality yielded

a p-value of < 0.0001. Figure 1 depicts a nonparametric density estimate of the these

measurements. We used a kernel density estimator with Gaussian kernel and bandwidth

choice by cross-validation. The distribution is apparently non-normal and asymmetric,

having several modes. The bump around 175 is not an artifact and also present in the

larger flasher list. For the present sample a mixture of normal distributions provides a rel-

atively reasonable approximation, although the bump around 175 would be ignored with

high probability when fitting such a model. Since according to expert knowledge and an-

alyzes of other data sets there is no ’typical shape’ for such data, it is better to use a

nonparametric approach which avoids specific assumptions on the shape.

180 190 200 210 220

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Figure 1.— Kernel density estimate of the flasher list measurements.


Applying our procedure with nominal error rates 1 − α = β = 0.05, AQL pα = 0.01

and RQL pβ = 0.05, we obtained n = 29 and c = 10.82. Thus, a random sample of 29

modules was drawn and analyzed yielding X ′29 = 216.30. The decision rule was applied

with τ = µ0 − 0.1 · µ0, where µ0 = 220 specifies the nominal power output stated by the

producer. Since T29 = 9.452, the lot is rejected.

4.2. Small sample properties

To get some insight into the statistical properties, particularly the dispersion and the

distributional shape of the estimates nm and cm, we performed Monte Carlo experiments.

Having in mind the situation of the data analysis above, we selected simulation models with

realistic means and variances, although, of course, the estimates nm and cm do not depend

on location and scale. Model 1 assumes that the measurements Xi are normally distributed

with mean 220 and variance 4. The other two models assume mixture distributions. Under

model 2,

Xi ∼ F2 = 0.1N(210, 6) + 0.9N(230, 4),

whereas under model 3

Xi ∼ F3 = 0.9N(220, 4) + 0.1N(230, 8).

Note that F2 is skewed to the left and F3 is skewed to the right. Further, F−12 (pβ)−F−1

2 (pα)

is larger than F−13 (pβ) − F−1

3 (pα). Thus, we expect smaller sample sizes under model 2

than under model 3.

To investigate the finite sample behavior under the above models we considered E(nm),

sd(nm), the quartiles q0.25, q0.5, q0.75 of the distribution of nm, E(cm), and sd(cm). The

sample size m of the additional data was chosen as m ∈ 100, 250, 500, 50000. AQL and

RQL were fixed at pα = 0.02 and pβ = 0.05, respectively, and the nominal error probabili-

ties were chosen as β = 1− α = 0.05. The following table provides Monte-Carlo estimates

based on 50000 repetitions.


Model m E(nm) sd(nm) q0.25 q0.5 q0.75 E(cm) sd(cm)

1 100 299.35 17.3 51 106 239 23.13 18.86

1 250 108.73 10.43 50 79 129 17.47 6.56

1 500 83.72 9.15 52 72 101 16.11 4.05

1 50000 65.39 8.09 63 65 68 14.89 0.35

2 100 88.57 9.41 12 28 66 17.85 16.55

2 250 35.35 5.95 16 25 42 14.74 6.08

2 500 28.65 5.35 17 24 35 14.06 3.82

2 50000 23.45 4.84 23 23 24 13.37 0.34

3 100 614.14 24.78 86 178 414 25.37 25.52

3 250 192.55 13.88 86 137 227 19.07 7.37

3 500 147.07 12.13 90 125 178 17.57 4.45

3 50000 114.09 10.68 110 114 118 16.19 0.38

The quantiles indicate that the distribution of nm is quite skewed for small to moderate

sample sizes m, and, depending on the true distribution, its dispersion can be quite large.

Although still a bit unpleasant, for larger sample sizes of the flasher list (as, e.g., m = 500)

the skewness is no longer a severe problem, since it implies conservative sample sizes.

The question arises how the above findings depend on the chosen values for AQL and

RQL, i.e., on the parameter α. Figure 2 illustrates for model 3 in terms of the 10% and

90% quantiles how the distribution of n500 depends on α = 1− β ∈ [0.9, 0.99].

To summarize, the above simulations support the applicability of our proposal and the

assertion of the asymptotic results, although the method is not practical in all cases. An

investigation of, e.g., a modified rule or a procedure employing accompanying rules is

beyond the scope of the present article.

4.3. Investigation of the moment condition

The above results provide some insight into the behavior of our procedure for small to

moderate sample sizes, if the assumptions on the underlying distribution are satisfied. But

what does happen if they are not satisfied? To shed some light on that issue we investigated

to which extent the fourth moment condition of Assumption 2.1 is really required. For that

purpose we simulated measurements following a symmetric Pareto type distribution, i.e.,


0.90 0.92 0.94 0.96 0.98

010

020

030

040

050

0

Figure 2.— Median (dashed line), 10%-, and 90%-quantiles of the distribution of nm

as a function of α = 1− β.

for γ > 0 we put

Gγ(x) =

1− 12(1 + x)−γ, x ≥ 0,

12(1− x)−γ, x < 0,

to define the location-scale family F(µ,σ2),γ(x) = Gγ((x − µ)/σ), µ ∈ R, σ > 0. As is well

known, the rth moment of Gγ exists for all 0 < r < γ, but the γth moment does not. Since

we are interested now on the large sample (asymptotic) distribution, we put m = 20000. As

in our first analysis, AQL and RQL were chosen as pα = 0.02 and pβ = 0.05, respectively,

and the nominal error probabilities are given by β = 1 − α = 0.05. For fixed γ > 0

the distance between the empirical distribution function of the standardized simulated

nm values, denoted by Hγ, and the N(0, 1) distribution function was measured by the

Kolmogorov-Smirnov statistic. This means, for S simulated values nm(1), . . . , nm(S), we

calculated

Dγ = supx|Hγ(x)− Φ(x)|.


where

Hγ(x) =1

S

S∑i=1

1(−∞,x]((nm(i)− nm)/sm), x ∈ R,

with nm = S−1∑Si=1 nm(i) and s2

m = (S − 1)−1∑Si=1(nm(i) − nm)2. Notice that Dγ

consistently estimates Dγ = ‖Hγ − Φ‖∞, where Hγ stands for the true distribution of

(nm − E(nm))/(V(nm))1/2. For each γ the Monte Carlo estimate Dγ was based on S =

500000 i.i.d. replications, independent from each other.

If the existence of the fourth moment is required for the theory to hold, Dγ should be

small for γ > 4 and get large as γ decreases. Figure 3 depicts Dγ for γ = 2.0, 2.1, . . . , 4.5.

Since there is still some estimation error and to support visual evaluation, we added a

Nadaraya-Watson kernel smooth of the simulated values using a bandwidth 0.5. The

result indicates that there is no hope to weaken Assumption 2.1.

2.0 2.5 3.0 3.5 4.0 4.5

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3.— Kolmogorov-Smirnov distance, Dγ, between the empirical distribution of

nm and the d.f. of the standard normal distribution under a Pareto type distribution, Gγ,

for γ = 2.0, 2.1, . . . , 4.5.


5. PROOFS OF MAIN RESULTS

The crux of the proofs of Theorems 3.1 and 3.2 (which will be carried out in Subsections

5.1 and 5.2, respectively) is Lemma 5.2 below. The proof of this lemma relies on the

functional delta method (which is recalled in the Appendix C), and on Lemma 5.1 (resp.

Corollary 5.1) for which we need some notation. We write D(R) for the space of all cadlag

functions on R. Moreover, we adopt Dudley’s notion of convergence in distribution of

random elements in D(R) with respect to the uniform metric (see, for instance, [9, Chapter

V]). In particular, we regard D(R) as a measurable space with respect to the σ-algebra

generated by all finite-dimensional projections (or, equivalently, by the ‖.‖∞-closed balls).

We also equip D(R) with the supremum norm ‖.‖∞ to make it a normed space. For the

considerations in this section we may assume without loss of generality ∆ = 0 in (2.10)

and (2.11). In particular, we may and do write for the sake of clarity Xi instead of X∆i .

We refer to the transpose of an Euclidean vector v as v′, and we set, for t ∈ R,

Fm(t) :=1

m

m∑i=1

1(−∞,t](Xi) and Fm(t) := G

(t− Xm

σm

).

To simplify the exposition we will sometimes refer to the distribution function F(µ,σ2) of

the Xi under P(µ,σ2) simply as F .

Lemma 5.1 Suppose Assumptions 2.1-2.2 hold. Then, under P(µ,σ2), as m→∞,

(5.1)√m(Fm − Fm

)d−→ BF (in D(R)).

Here BF (.)d= BF (.) − [F(µ,σ2)(.)](ξµ, ξσ2)′, where F(µ,σ2) is defined as in (2.2), BF is an

F -Brownian bridge, and ξµ, ξσ2 and BF are jointly normal with 0 mean and

Cov(µ,σ2)(ξθ, ξθ′) = Cov(µ,σ2) (ψθ(X1), ψθ′(X1))

Cov(µ,σ2)(ξθ, BF (t)) = Cov(µ,σ2)

(ψθ(X1),1(−∞,t](X1)

)for θ, θ′ ∈ µ, σ2, ψµ(x) := x− µ and ψσ2(x) := (x− µ)2 − σ2.

On an informal level, BF can be seen as an “F -Brownian bridge with drift”. Note that

this process is continuous since BF and [F(µ,σ2)(.)](ξµ, ξσ2)′ are; recall that F and F(µ,σ2)

are continuous. In particular, it actually does not matter whether we consider convergence

in distribution w.r.t. the uniform metric or w.r.t. the Skorohod metric (cf. [1, p.110]). We

imposed the supremum metric for the simple reason that we intend to apply the functional

delta method later on.


Remark 5.1 It is straightforward to check that the covariance function, Γ(µ,σ2), of the

Gaussian process BF in Lemma 5.1 is given by

Γ(µ,σ2)(s, t) = F(µ,σ2)(s)(1− F(µ,σ2)(t)

)− F(µ,σ2)(t)b(µ,σ2)(s)(5.2)

−F(µ,σ2)(s)b(µ,σ2)(t) + F(µ,σ2)(s)A(µ,σ2)F(µ,σ2)(t)′

for s ≤ t, where F(µ,σ2) is defined as in (2.2), and

b(µ,σ2)(t) :=

E(µ,σ2)

[(X1 − µ)1(−∞,t](X1)

]E(µ,σ2)

[((X1 − µ)2 − σ2) 1(−∞,t](X1)

] ,

A(µ,σ2) :=

σ2 E(µ,σ2)[(X1 − µ)3]

E(µ,σ2)[(X1 − µ)3] E(µ,σ2)[(X1 − µ)4]− σ4

.

Proof: (of Lemma 5.1) The proof of Lemma 5.1 is an application of a result which

is mainly associated with Darling and Durbin, cf. [9, Example V.15], or [17, Theorem

19.23]. According to that, it suffices to show that the mapping (µ, σ2) 7→ F(µ,σ2) is Frechet

differentiable at (µ, σ2), and that

(5.3)√m

Xm − µσ2m − σ2

=1√m

m∑i=1

ψµ(Xi)

ψσ2(Xi)

+ oP(µ,σ2)(1)

holds for some functions ψµ and ψσ2 satisfying E(µ,σ2)[ψµ(X1)] = 0, E(µ,σ2)[ψσ2(X1)] = 0

and E(µ,σ2)

[∑θ∈µ,σ2 ψ

2θ(X1)

]< ∞. Since the fourth moment of G is assumed to be

finite (Assumption 2.1), we may apply the central limit theorem to the random variables

(Xi − µ)2, i = 1, 2, . . . . In view of this, it is easy to see that (5.3) is fulfilled for ψµ and

ψσ2 defined as in the statement of Lemma 5.1. The Frechet differentiability is ensured by

Assumption 2.2. Q.E.D.

Corollary 5.1 Suppose Assumptions 2.1-2.2 hold, and set η(t) := σt+µ. Then, under

P(µ,σ2), as m→∞,

(5.4)√m(Gm −G

)d−→ BF η

d= BG (in D(R)),

where BF is as in Lemma 5.1. BG is a centered Gaussian processes with covariance function

given by (5.2) with (µ, σ2) = (0, 1), i.e., by Γ(0,1) = Γ from (??).

Proof: We denote by C(R) the space of strictly increasing R-valued continuous functions

ψ on R with ψ(−∞) = −∞ and ψ(+∞) = +∞. We equip C(R) with the Borel-σ-algebra


C related to the metric d∞(ψ1, ψ2) :=∑∞k=1 2−k min1, supx∈[−k,k] |ψ1(x) − ψ2(x)|. The

process ηm(t) := σmt + Xm clearly converges in probability, and therefore in distribution,

(in C(R)) to the deterministic process η. Since BF is continuous, the convergence in (5.1)

holds in particular w.r.t. the Borel-σ-algebra, D, related to the Skorohod metric. Hence we

may apply the theory of [5]. Firstly, Theorem 4.4 of [5] yields convergence in distribution of

(√m (Fm− Fm), ηm) to (BF , η) on D×C. Note that the product σ-algebra D×C coincides

with the Borel-σ-algebra related to d((φ1, ψ1), (φ2, ψ2)) := max‖φ1 − φ2‖∞, d∞(ψ1, ψ2).The map (φ, ψ) 7→ φψ from D(R)×C(R) to D(R) is easily seen to be (d, ‖.‖∞)-continuous

at each point of C(R)× C(R) (we write C(R) for the space of continuous functions on R).

Since moreover (BF , η) is concentrated on C(R)× C(R), Corollary 1 to Theorem 5.1 of [5]

shows that√m (Fm − Fm) ηm

d→ BF η on D. The limit BF η is of course continuous,

and therefore the convergence in distribution holds w.r.t. the uniform metric too. But

this proves the convergence in distribution to BF η in (5.4) since Gm = Fm ηm and

G = Fm ηm. It remains to show BF ηd= BG. For s ≤ t we obviously have

Cov(µ,σ2)(BG(s), BG(t)) = Cov(µ,σ2)(B

F η(s), BF η(t)) = Γ(µ,σ2)(η(s), η(t)).

Then straightforward calculations, using E(µ,σ2)[(X1 − µ)k] = σkE(0,1)[Xk1 ], show that this

expression coincides with Γ(0,1)(s, t). Q.E.D.

Lemma 5.2 Suppose Assumptions 2.1-2.3 hold. Then, for every k ∈ N, p1, . . . , pk ∈ (0, 1)

and λ1, . . . , λk ∈ R,

√m

k∑i=1

λi(G−1m (pi)−G−1(pi)

)d−→

k∑i=1

λiBG(G−1(pi))

G′(G−1(pi))(in R)

under P(µ,σ2), as m→∞, where BG is as in Corollary 5.1.

Proof: Let D be the subset of D(R) that consists of all non-decreasing functions φ

satisfying φ(−∞) = 0 and φ(+∞) = 1. Moreover, let D0 be the subset of D(R) that

consists of all functions φ being continuous at φ−1(p), and let Qp : D → R be defined by

Qp(φ) := φ−1(p). Because of 0 < G′(G−1(p)) <∞ (by Assumption 2.3) and the continuity

of G, it can be shown (with the help of the Arzela-Ascoli theorem) that Qp is Hadamard

differentiable at G tangentially to D0 with Hadamard derivative

DHadG;D0

Qp (φ) =φ(G−1(p))

G′(G−1(p))(φ ∈ D0).

This is more or less standard (cf. [1, Proposition II.8.4], for example), and therefore we

omit the details. As an immediate consequence we obtain

(5.5) DHadG;D0

Q (φ) =k∑i=1

λiφ(G−1(pi))

G′(G−1(pi))(φ ∈ D0)


for Q : D → R defined by Q(φ) :=∑ki=1 λiQpi(φ). Because of Corollary 5.1 and (5.5), we

can apply the functional Delta method (cf. the Appendix C) to obtain

(5.6)√m

k∑i=1

λi(G−1m (pi)−G−1(pi)

)d−→ DHad

G;D0Q (BG) (in R)

as m→∞. By virtue of (5.5), this proves Lemma 5.2. Q.E.D.

5.1. Proof of Theorem 3.1

We are now in the position to finish the proof of Theorem 3.1 without further obstacles.

By (1.1) we have p = P(µ,σ2)[X1 ≤ τ ], and therefore

(5.7)τ − µσ

= G−1(p).

Now, Tn > c holds if and only if

√nX ′n − µσ

+√n(X ′n − µσm

− X ′n − µσ

)> c+

√n G−1

m (p) +√n(τ − µσm

−G−1m (p)

)

which in turn is equivalent to√nX ′n − µσ

+√nX ′n − µσ

σ − σmσm

(5.8)

+√n(G−1m (p)−G−1(p)

)+√n(G−1(p)− τ − µ

σm

)> c+

√n G−1(p).

We denote the four summands on the left-hand side of (5.8) by S1(n), . . . , S4(n). The

first one is asymptotically standard normal by the central limit theorem. Because σ−σmσm

converges P(µ,σ2)-almost surely to σ−σσ

= 0 by the strong law of large numbers, S2(n)

converges in probability to 0 by Slutsky’s lemma. According to Lemma 5.2 and (2.12),

S3(n) also converges in probability to 0 by Slutsky’s lemma. We further obtain by (5.7),

S4(n) =√n(G−1(p)−G−1(p)

σ

σm

)=G−1(p)

σm

√n√m

√m (σm − σ).

Now,√m(σm − σ) can be split into the sum of U1(m) :=

√m(σm − σm) and U2(m) :=

√m(σm − σ), where σm := ( 1

m

∑mi=1(Xi − µ)2)1/2. The summand U2(m) is asymptotically

normal with 0 mean and variance Var(µ,σ2)[(X1 − µ)2]. Multiplying by√n/√m yields


convergence in probability to 0, since we assumed (2.12). For m ≥ 2 we also obtain

|U1(m)| =√m

∣∣∣∣∣(

1

m− 1

m∑i=1

(Xi − Xm)2)1/2

−(

1

m− 1

m∑i=1

(Xi − µ)2)1/2

∣∣∣∣∣≤√m

√m√

m− 1

∣∣∣∣∣ 1

m

m∑i=1

(Xi − Xm)2 − 1

m

m∑i=1

(Xi − µ)2

∣∣∣∣∣1/2

≤√m√

2

∣∣∣∣∣ 1

m

m∑i=1

(− X2

m + 2Xiµ− µ2)∣∣∣∣∣

1/2

=√

2√m

∣∣∣−X2m + 2Xmµ− µ2

∣∣∣1/2=√

2√m∣∣∣Xm − µ

∣∣∣.Thus, since we assumed (2.12) and since

√m(Xm − µ) is asymptotically N(0, σ2), we

deduce with the help of Slutsky’s lemma that (√n/√m)|U1(m)| converges in probability

to 0. Moreover, G−1(p)/σm converges almost surely to G−1(p)/σ, so that Slutsky’s lemma

ensures that S4(n) converges in probability to 0. This completes the proof of Theorem 3.1.

5.2. Proof of Theorem 3.2

The proof of Theorem 3.2 relies on the delta method for random variables (cf. the Ap-

pendix C) and Lemma 5.2. Indeed, Lemma 5.2 implies

√m(G−1m (pβ)−G−1

m (pα)− (G−1(pβ)−G−1(pα)))

d−→ BG(G−1(pβ))

G′(G−1(pβ))− BG(G−1(pα))

G′(G−1(pα)).

Now, the left-hand side of (3.8) can be expressed as

√m(f(G−1

m (pβ)−G−1m (pα))− f(G−1(pβ)−G−1(pα))

)with f(x) = (Φ−1(1 − α) − Φ−1(1 − β))2x−2, recall (3.6). Therefore, Theorem 3.1 of [17]

shows that the left-hand side of (3.8) converges weakly to the law of the normal random

variable

−2(Φ−1(1− α)− Φ−1(1− β))2

(G−1(pβ)−G−1(pα))3

(BG(G−1(pβ))

G′(G−1(pβ))− BG(G−1(pα))

G′(G−1(pα))

)

since f ′(x) = −2(Φ−1(1 − α) − Φ−1(1 − β))2x−3. Now the proof can be completed easily

since we know the covariance function, Γ(0,1) = Γ, of BG from Corollary 5.1.

APPENDIX A: PROOF OF (2.3)

This appendix is devoted to a proof of the sufficiency of condition (2.3) (along with

smoothness of G) for (2.1). This result is quite interesting on its own. A similar analysis

in a more general setting can be found in [15].


Lemma A.1 If G is twice continuously differentiable and satisfies (2.3), then (2.1) holds.

Proof: Let f(µ,σ2) and g denote the Lebesgue densities of F(µ,σ2) and G, respectively.

Since g is continuously differentiable and

(A.1) f(µ,σ2)(x) =1

σg(x− µσ

),

we obtain continuity of the mapping (σ2, x) 7→ ∂∂σ2f(µ,σ2)(x) restricted to σ2 > 0. Together

with the integrability of f(µ,σ2), this justifies the following interchange of the derivative and

the integral,

∂

∂σ2F(µ,σ2)(t) =

∂

∂σ2

∫ t

−∞f(µ,σ2)(x)dx =

∫ t

−∞

∂

∂σ2f(µ,σ2)(x)dx.

In the same way we obtain the analogue for “ ∂∂µ

”. With the help of (A.1) we deduce for

F(µ,σ2), defined in (2.2),

F(µ,σ2)(t) =

(− 1

σ2

∫ t

−∞g′(x− µσ

)dx, − 1

σ2

∫ t

−∞

x− µ2σ2

g′(x− µσ

)dx

).

Thus, (2.1) follows if we can prove that

(A.2)∫ t

−∞

1

|v|

∣∣∣∣∣ 1σ g(x− (µ+ v1)√

σ2 + v2

)− 1

σg(x− µσ

)− 1

σ2

(v1 +v2

x− µ2σ2

)g′(x− µσ

)∣∣∣∣∣dx→ 0

uniformly in t, as |v| → 0, where v = (v1, v2)′. In the remainder of the proof we will

establish (A.2).

Let ε > 0. We split the integral in (A.2) into∫(−∞,t]

· · · =∫

(−∞,t]∩[aε,bε]c· · ·+

∫(−∞,t]∩[aε,bε]

. . . =: Sε1(t, v) + Sε2(t, v)

for some real numbers aε, bε satisfying aε < bε. We intend to prove (A.2) by showing that

both Sε1(t, v) and Sε1(t, v) are bounded by ε/2 uniformly in t for sufficiently small |v|, for a

suitable choice of aε and bε. To estimate Sε1(t, v) we note that, by the Mean Value Theorem,

we have for some ξ inbetween (x− µ)/σ and (x− (µ+ v1))/√σ2 + v2,

(A.3)1

σg(x− (µ+ v1)√

σ2 + v2

)− 1

σg(x− µσ

)=(x− (µ+ v1)√

σ2 + v2

− x− µσ

)1

σg′(ξ).


Hence the integrand in (A.2) is bounded by∣∣∣∣x−(µ+v1)√σ2+v2

− x−µσ

∣∣∣∣√v2

1 + v22

∣∣∣∣ 1σg′(ξ)∣∣∣∣ +

∣∣∣v1 + v2x−µ2σ2

∣∣∣√v2

1 + v22

∣∣∣∣ 1

σ2g′(x− µσ

)∣∣∣∣≤

∣∣∣∣∣∣xσ − (µ+ v1)σ − x√σ2 + v2 + µ

√σ2 + v2

σ√σ2 + v2

√v2

1 + v22

∣∣∣∣∣∣∣∣∣∣ 1σg′(ξ)

∣∣∣∣ +∣∣∣∣1 +

x− µ2σ2

∣∣∣∣ ∣∣∣∣ 1

σ2g′(x− µσ

)∣∣∣∣=

1

σ2

∣∣∣(x− µ)(σ −√σ2 + v2)− v1σ

∣∣∣√σ2 + v2

√v2

1 + v22

|g′(ξ)| +1

σ2

∣∣∣∣1 +x− µ2σ2

∣∣∣∣ ∣∣∣∣g′(x− µσ)∣∣∣∣

≤ 1

σ2

|x− µ| · |v2| 1

2√

minσ2,(σ2+v2)+ |v1|σ

√σ2 + v2

√v2

1 + v22

|g′(ξ)| +1

σ2

∣∣∣∣1 +x− µ2σ2

∣∣∣∣ ∣∣∣∣g′(x− µσ)∣∣∣∣ .

For the latter step we used the inequality

(A.4) |√a−√b| =

∣∣∣∣ a− b√a+√b

∣∣∣∣ ≤ |a− b|2√

mina, b(a, b ≥ 0)

to estimate the expression |σ −√σ2 + v2| = |

√σ2 −

√σ2 + v2|. Now it follows easily that

the integrand in (A.2) is bounded by

Cµ,σ (1 + |x|)(|g′(ξ)| +

∣∣∣∣g′(x− µσ)∣∣∣∣)

for some constant Cµ,σ > 0 that may depend on µ and σ, provided |v| is so small so that

(σ2 + v2) is larger than (for instance) σ2/2. Since g′ = G′′, we deduce with the help of our

basic assumption (2.3) that the integral in (A.2) with t =∞ is finite. Because of this and

the continuity of the integrand, we may choose aε and bε in such a way that Sε1(t, v) < ε/2

for all t ∈ R (and sufficiently small |v|).

It remains to show Sε2(t, v) < ε/2 for all t ∈ R (and sufficiently small |v|). With the help

of (A.3) we obtain as before that the integrand in (A.2) equals∣∣∣∣∣∣∣x−(µ+v1)√

σ2+v2− x−µ

σ√v2

1 + v22

1

σg′(ξ) −

(v1 + v2

x−µ2σ2

)√v2

1 + v22

1

σ2g′(x− µσ

)∣∣∣∣∣∣∣ .By the triangle inequality this expression is bounded above by∣∣∣∣∣∣∣

x−(µ+v1)√σ2+v2

− x−µσ√

v21 + v2

2

−

(v1 + v2

x−µ2σ2

)√v2

1 + v22

1

σ2g′(ξ)∣∣∣∣∣∣∣

+

∣∣∣∣∣∣(v1 + v2

x−µ2σ2

)√v2

1 + v22

1

σ2g′(ξ)−

(v1 + v2

x−µ2σ2

)√v2

1 + v22

1

σ2g′(ξ)∣∣∣∣∣∣ .


Doing some elementary manipulations, we obtain the upper bound|x− µ|∣∣∣∣∣∣∣∣∣

√σ2+v2−

√σ2√

σ2+v2− v2

2σ2√v2

1 + v22

∣∣∣∣∣∣∣∣∣+∣∣∣∣∣∣∣∣v1

(σ2√σ2+v2

− 1)

√v2

1 + v22

∣∣∣∣∣∣∣∣ 1

σg′(ξ)(A.5)

+

∣∣∣∣∣∣(v1 + v2

x−µ2σ2

)√v2

1 + v22

∣∣∣∣∣∣ 1

σ2

∣∣∣∣g′(x− µσ)− g′(ξ)

∣∣∣∣=:

(|x− µ||U1(v)|+ |U2(v)|

)1

σg′(ξ) + |U3(v)| 1

σ2

∣∣∣∣g′(x− µσ)− g′(ξ)

∣∣∣∣ .Now,

U1(v) =

√σ2 + v2 −

√σ2

v2

1√σ2 + v2

− 1

2σ2+ o|v|(1)

≤v2

1

2 min√σ2,σ2+v2

v2

1√σ2 + v2

− 1

2σ2+ o|v|(1),

where we used (A.4). Thus we have U1(v)→ 0 as |v| → 0. It is easy to see that also U2(v)→0 as |v| → 0. In both cases the convergence holds uniformly in t and x. Moreover, U3(v) is

bounded uniformly in t and x, for sufficiently small |v|. As g′ is continuous, it is uniformly

continuous on [aε, bε] (aε, bε have been fixed above). Therefore |g′((x − µ)/σ) − g′(ξ)|converges to 0 uniformly in t ∈ R and x ∈ [aε, bε], as |v| → 0. Altogether, the expression

in (A.5) converges to 0 uniformly in t ∈ R and x ∈ [aε, bε], as |v| → 0. Hence Sε2(t, v) is

indeed smaller than ε/2 for all t ∈ R (and sufficiently small |v|). Q.E.D.

APPENDIX B: OPTIMAL SAMPLING PLAN

We shall now provide the details of the following fact. If (2.7) is plugged into (2.5)-(2.6),

the optimal sampling plan (n, c), i.e., the sampling plan with the minimal n, satisfying

(2.5)-(2.6) is given by (2.8)-(2.9). Since the right-hand side of (2.7) is strictly decreasing

in p, the requirements (2.5)-(2.6) are equivalent to

(B.1) Φ−1(1− α) ≥ c+√n G−1(pα),

(B.2) Φ−1(1− β) ≤ c+√n G−1(pβ).

It is easily seen that (n, c) with the minimal n satisfying (B.1)-(B.2) is given by the intersec-

tion of the mappings n 7→ (Φ−1(1−α)−√nG−1(pα)) and n 7→ (Φ−1(1−β)−

√nG−1(pβ)),

i.e., characterized by

(B.3) Φ−1(1− α)−√nG−1(pα) = Φ−1(1− β)−

√n G−1(pβ).


This equation has a solution n since G−1(pα) is strictly smaller than G−1(pβ) (recall that

G is strictly increasing). Plugging this n into (B.3) we obtain the corresponding c. Hence

the optimal sampling plan (n, c) is indeed given by (2.8)-(2.9).

The derivation of the sampling plan (3.6)-(3.7) from (3.5) can be done along the same

lines. Actually, this requires that the right-hand side of (3.5) is strictly decreasing (i.e.,

that the map p 7→ Gm(p) is strictly increasing) and that G−1m (pα) < G−1

m (pβ). Since

the map p 7→ Gm(p) is a step function, the first requirement is violated and the second

requirement may be violated. In particular, the equivalence of (2.5)-(2.6) to (B.1)-(B.2)

(with G replaced by Gm) may be not anymore true, and the denominator of the right-

hand side of (3.6) may equal zero. On the other hand, Gm uniformly converges almost

surely to G (a Glivenko-Cantelli argument applies), and G−1m (p) converges almost surely to

G−1(p) for every p at which G is continuous and strictly increasing (cf., for instance, [17,

Section 21]). Therefore the sampling plan (3.6)-(3.7) is at least asymptotically optimal

and well-defined.

APPENDIX C: THE FUNCTIONAL DELTA METHOD

Here we briefly recall the essence of the functional delta method. For a more comprehen-

sive exposition see, e.g., [11, 17]. Various applications of the delta methods can be found

in [6, 13, 14] and in references cited therein. Let (D, ‖.‖D) and (E, ‖.‖E) be normed linear

spaces, and Df ,D0 ⊂ D and θ ∈ Df . Suppose (Tn) is a sequence of Df -valued random

elements and T is a D0-valued random element. Assume we know√n(Tn−θ)

d→ T , but we

are actually interested in the limit in distribution of√n(f(Tn) − f(θ)) for some function

f : Df → E. If D = E = R and f is differentiable with (Frechet) derivative f ′, then one

can use the Taylor expansion of order one to obtain√n(f(Tn)− f(θ)) =

√n (f ′(θ)(Tn − θ) + o(Tn − θ))

≈ f ′(θ)√n(Tn − θ),

and one would guess that this expression converges in distribution to f ′(θ)T . The delta

method ([17, Theorem 3.1]) shows that this is in fact true. This method can be made

rigorous also in the setting of general normed linear spaces D and E. Then, of course,

f ′(θ)(.) has to be replaced suitably. It turns out that the Hadamard derivative DHadθ;D0

f (.)

of f at θ tangentially to D0 is the right analogue of f ′(θ)(.) (note that in finite-dimensional

spaces the Hadamard and Frechet derivative coincide). In fact, provided the Hadamard

derivative exists, the functional delta method ([17, Theorem 20.8]) implies

√n(f(Tn)− f(θ))

d→ DHadθ;D0

f (T ).


For the notion of Hadamard differentiability see, for instance, [17, Section 20.2] or [6].

ACKNOWLEDGEMENT

We thank Dr. Werner Herrmann, TUV Rheinland Immissionsschutz und Energiesysteme

GmbH, Cologne, for facing us with the practical need of improved nonparametric meth-

ods for acceptance sampling and providing real data. The authors and Dr. Herrmann

acknowledge the financial support from the Solarenergieforderverein Bayern e.V. (SeV),

project ’Stichprobenbasierte Labormessungen’, and thank Prof. Becker (SeV) and Fabian

Flade (SeV). The constructive comments of an anonymous referee, which improved the

presentation of the results, are also acknowledged.

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